Let \(Q_i=(d_{\chi \psi })\)  be the decomposition matrix of \(B_i\)  for \(i\in \lbrace 1,2\rbrace \) . Then \(C_i:=Q_i^\text{t}Q_i\)  is the Cartan matrix of \(B_i\)  where \(Q_i^\text{t}\)  denotes the transpose of \(Q_i\) . By [1]} we have \(C_i^{-1}=\big ([\varphi ,\theta ]^0\big )_{\varphi ,\theta \in \operatorname{IBr}(B_i)}\) . It follows that
\(M_i=\bigg (\frac{1}{|G|}\sum _{g\in G_i^0}\sum _{\varphi ,\theta \in \operatorname{IBr}(B_i)}d_{\chi \varphi }d_{\psi \theta }\varphi (g)\theta \big (g^{-1}\big )\bigg )_{\chi ,\psi \in \operatorname{Irr}(B_i)}\nonumber \\\hphantom{M_i}{} =\bigg (\sum _{\varphi ,\theta \in \operatorname{IBr}(B_i)}d_{\chi \varphi }d_{\psi \theta }[\varphi ,\theta ]^0\bigg )_{\chi ,\psi \in \operatorname{Irr}(B_i)}=Q_iC_i^{-1}Q_i^\text{t}.\) 
